I've been trying out the new class Seraph. I quickly began to adore her character design including that main outfit. However, there's an issue with the skills. I cannot drag a single skill to the quickslots. I've reached lv 56 by doing the eternal winter storyline, yet no skills are available.

I rarely play other classes besides Tamer, but never have I encountered such issue with the other classes i've played either. Is this an issue specific to Seraph? I got a bit attached to this new class but it might not be worth the time and effort to progress like this. (I only play on the American server, since i ended up there when i returned to the game years ago, but it should not make any difference.)

I believe its less of an "issue" and more of a design philosophy. Both seraph and deadeye (the class released before) have very few skills that can be quickslotted. In case of seraph, this may be because of how strongly she is based on combos and skills that flow into each other, rather than singular skills that can be used via quickslot. 

In general, classes in BDO perform much better when using their skills through key combinations rather than quickslots, due to several limitations with quickslots (such as not all skills being available, and quickslot casting not being able to cancel previous skills), so I would recommend learning classes with the respecitve key combos for their skills. 

Trying to get the game to register multiple keypresses at the same time can be bothersome, making quickslots useful; also 1 keypress is less APM than multiple, which helps prevent carpal tunnel. Desync can disrupt combos so I'd rather not rely on them. Also, designing grindspots around the assumption that you full combo all the time can feel like a slog.

She already needs re-worked due to her BSR regan is a requirment to play her. That takes up light stones and soverign wep slots. She's cluttered with a disability implanted by devs for only endgame players being able to use her correctly.  
Adding since ive played her i really do wish they would add her awkaening as her base instead of bsr, add a chainsaw to it and go from there. But with heidal ball we should see some developer magic.